Abstract:

A prime number $p$ is called $b$-elite if only finitely many generalized Fermat numbers $F_{b,n}=b^{2^n}+1$ are quadratic residues modulo $p$. Let $p$ be a prime. Write $p-1=2^rh$ with $r\geq 0$ and $h$ odd. Define the length of the b-Fermat period of $p$ to be the minimal natural number $L$ such that $F_{b,r+L}\equiv F_{b,r} (\bmod p).$ Recently Müller and Reinhart derived three conjectures on $b$-elite primes, two of them being the following. (1) For every natural number $b>1$ there is a $b$-elite prime. (2) There are generalized elite primes with elite periods of arbitrarily large lengths. We extend Müller and Reinhart's observations and computational results to further support above two conjectures. We show that Conjecture 1 is true for $b\leq10^{13}$ and that for every possible length $1\leq L\leq40$ there actually exists a generalized elite prime with elite period length $L$.